No debt ceiling increase means the budget is balanced? Try telling your credit card company you’re not going to pay the tab you’ve run up, and see if they think your budget is balanced.
I think that Mike is saying that either would have the same immediate effect. If a balanced budget amendment was passed, then the government expenditures could not receive its revenues and thus no debt would be taken on. If the debt ceiling is not raised, then no additional debt could be taken on.
Of course, past that it would be quite different. The balanced budget amendment would ensure that no additional debt could ever be incurred (excluding whatever exceptions are written into the amendment). However, if the debt ceiling is not raised, then the government could incur more debt once some existing debt is paid off.